Question: Simplify the following expression and state the condition under which the simplification is valid: $k = \dfrac{x^2 + 6x}{x^2 - x - 42}$
Answer: First factor the expressions in the numerator and denominator. $ \dfrac{x^2 + 6x}{x^2 - x - 42} = \dfrac{(x)(x + 6)}{(x - 7)(x + 6)} $ Notice that the term $(x + 6)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(x + 6)$ gives: $k = \dfrac{x}{x - 7}$ Since we divided by $(x + 6)$, $x \neq -6$. $k = \dfrac{x}{x - 7}; \space x \neq -6$